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More separation axioms
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A $X$ is completely regular if $X$ is $T_1$ and if  $x_0 \in X$, $C$  closed in $X$, 
 and $x_0 \not\in 
C$, then there exists a continuous function $f: X \rightarrow [0, 1]$ such that $f(x_0) = 1$ and $f(C) 
= \{0\}$
\v
Ex:  ${\cal R}_{\cal L}$ is completely regular.

If for any $C, D$ disjoint closed sets in $X$, there exists a continuous function $f: X \rightarrow
[0, 1]$ such that $f( C) = \{0\}$ and $f(D) = \{1\}$, then $X$ is normal.

Urysohn Lemma:  If $X$ is normal then for any $C, D$ disjoint closed sets in $X$, there exists a
continuous function $f: X \rightarrow [0, 1]$ such that $f( C) = \{0\}$ and $f(D)  = \{1\}$

$X$ is completely normal if every subspace of $X$ is normal.


Lemma 33.2:  A subspace of a $T_i$ space is $T_i$ for $i = 0, 1, 2, 3, 3.5$, but not $i = 4$
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A product  of  $T_i$ spaces is $T_i$ for $i = 0, 1, 2, 3, 3.5$, but not $i = 4$

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34:   2nd countable and regular implies metrizable.

35:  Tietze extension theorem:  Let $X$ be a normal space, and let $A$ be a closed subspace of $X$.
\v
a.)  Any continuous map $f: A \rightarrow [a, b]$, can be extended to a continuous map $f: 
X \rightarrow [a, b]$
\v
b.)  Any continuous map $f: A \rightarrow {\cal R}$, can be extended to a continuous map $f: 
X \rightarrow {\cal R}$




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